A function f(x) is said to be 'rational' if it is computable with a finite number of 'elementary operations' +,-,* and \... all other functions are said to be 'irrational'...

Thanks, this slipped my mind somehow.

Then are we allowed to use irrational constants to build rational functions? I.e., if a function is a ratio of two polynomials, can the polynomials' coefficients be irrational? If no, then the document provided above is still sufficient because it shows that for .

Then are we allowed to use irrational constants to build rational functions?... i.e., if a function is a ratio of two polynomials, can the polynomials' coefficients be irrational?...

From the 'theoretical' point of view, if You have a computer capable to menage irrational numbers [i.e. it has a memory of 'infinite dimension' and an infinite 'speed of computation'...], the ratio of two polynomials can be computed with a finite sequence of elementary operations...

I am asking whether rational functions can use irrational constants with respect to the OP's problem. If irrational numbers are not allowed, then the problem has been solved. Otherwise, more thought is needed, but I don't want to go there if it's not necessary.